Career: Identifying and Formalizing
Local Mechanisms of Cellular Organization
Project Description idea : a phenomenon is understood mechanistically when we can describe it locally. The phenomenon is fully understood when we can abstract that mechanism as a design principle. (For example, a tire, an and x and y aer all wheels,; d,f, ang are all levers.) If we are lucky, the mechanism can even be understood, with some equivalence, as a global mechanism. Since our universe is a local, mechanistic universe, whether all phenomenon can be explained using continuum methods is a philosophical one. Success in physics has resulted in a paradigm where it is believed that a complete understand means an equivalence with a glocal mechanism. However, real success in biology may turn this expectataion around, and we may revert to understanding that mechanistic understanding is true understaning, and in some cases all that can be expected. My goal is to push our understanding of when a mechanistic understanding can be gloablized further. idea : Biology is difficult to approach from a classical engineering or physics perspective due to aspects that are unique to biology: cells are individuals, all different and selfenergetic. However, biological cells do organize and are fairly successful in this. How do they do it? Throughout biology, there are certain solutions that are ubiquitouis. One of thes is the importance of alignment in coordinating populations. A cell receives and send information about its environment locally.
Introduction (1 page)
Biology is local (2 pages)
Myxobacteria (2 pages)
Limb (2 pages)
Prostate (2 pages)
Interdisciplinary Conference (1 page)
Education (2 pages)

Interactions in Biology Are Local (just like physics) (2 pages)
Modeling in biology requires the development of new techniques that reflect the uniqueness of biological systems. Success in biological modeling will likely spawn a new paradigm for our concept of what it means to understand a complex system. We live in a mechanistic, causal reality where every event B is preceded by an event A and a complete understanding of B means that we can predict B from A in a level of detail that is only limited by the resolution of our information about A. Given the velocity and position of 5 billiard balls on a pool table, we can predict all collisions and their final resting locations.
In physics, especially classical Newtonian physics, we have had enormous success in understanding and describing the local, mechanistic interactions with global, continuum theories of interaction. The diffusion of dye from a point source in water is Brownian motion, described in detail as both a local mechanism and continuum mechanism by
Einstein. However, despite its success as a description, diffusion is not a global mechanism. According to the heat equation, diffusion propagates at infinite speed. This is not physical. According to the heat equation, the source of the dye is a point and the dye itself is infinitesimally divisible. Again, these assumptions are not physical.
Are enormous success in describing physics with global mechanisms reflects certain characteristics of the subject of physics: molecules of dye diffusion in water are identical enough, and small enough, and energetically neutral enough, that there is not too much loss of information if they lose their individual identities. Each individual local interaction is not important.
In contrast, biological cells are not energetically neutral. Each biological cell has an individual purpose.
We have had a success explaining a large number of biological phenomena from the global, continuum perspective. Animal flocking, chemotaxis, : all phenomena in which the individuality of biological cells could be averaged away.
I am interested in biological phenomena in which the individuality of cells cannot be averaged away, in which one must begin with local mechanisms. I would like to create novel mathematics for these problems that are specifically designed and tailored for the biological problem in question. New approaches to biological problems will at first seem ad hoc. There was something unique about that biological phenomenon that made it nonamenable global methods. This unique aspect requires a novel local mechanism of interaction. Once the salient features of the new local mechanism phenomenon has been captured using whatever innovative methods necessary, I would like to formalize the local mechanism. What was essential, salient characteristic of that interaction? Can that interaction be abstracted and formalized? Can it be understood as and replaced with an approximate global mechanism?
In the several sections that follow, I will provide examples of how this has been accomplished. Many of these examples are my own work, because this philosophy has

been a result of my research and has developed over 10 years. In each case, describing local mechanisms involved novel methods, and in some cases, these local mechanisms could be described using a global formalism. I will also include questions where the local mechanisms have not been entirely formalized, where there is still interesting work to be done.
Many of these examples involve cellular alignment. Cellular alignment appears to be a universal mechanism of cellular organization in biolopgy. (list examples) Shaking a dozen pencils in a tray, it is intuitive that many of the pencils will align in parallel arrays.
This is a mechanistic result of the elongation of the pencils. In biology, cells are frequently elongated and the elongation of a cell is a strong indicator that that cell is coordinating with other cells. While the elongation results in spatial-angular coordination in the case of the energetically inert pencil, the perspective of the biological cell is that it is elongated so as to coordinate.

Myxobacteria Fruiting Body Formation (2 pages)
One non-biological characteristic of our fruiting bodies in simulations were the small size of our fruiting bodies. In nature, fruiting bodies are many orders of magnitude larger than the ones that resulted in our simulations.
A hypothesis for fruiting body size and a new direction for connecting bacterium persistence length with a concept of momentum : Fruiting bodies are hollow because myxobacteria cells, ever in motion, are resistant to sharp turns. They persist in their direction of motion and as such move in lines. These can be lines that extend indefinitely
(streams) or lines that turn in on themselves (torii). Thus it is the persistence length of the myxobacteria that should determine the radius of the fruiting body. I would like to verify this hypothesis.
In physics, inerta …momentum. To what extent can we think of the persistence length of a bacterium as analogous to momentum?
Moving Through An Energetic Landscape: Transitions Between Fruiting Body
Configurations Transitions are usually not smooth, but disordered. However, they result in particular patterns with statistical regularity. During the state of being disordered, is there any information? If the cells were placed down at random, do they behave the same? What are the mechanisms driving from one shape to another. Kevrekidis

Mammalian Bone Development (2 pages)
One non-biological characteristic of our fruiting bodies in simulations were the small size of our fruiting bodies. In nature, fruiting bodies are many orders of magnitude larger than the ones that resulted in our simulations.
Considering limbs across mammalian species, one of the most obvious features is the amount of variation among species, and interesting (and relatively frequent) variation within species (for example, 6 fingers is a dominant trait).
Having already developed a model that allows for stage-dependent bone formation, I would like to begin with a geometrically trivial rectangle in which to grow limbs. As the rectangle expands in the x and y directions, this should effect the number of limbs. I would like to develop a theory for inter-digitation and bone size and it’s dependence on the rate of growth of the rectangle. While I will not know how the limb growth varies from species to species, I will be able to describe what aspects of limb growth do vary across species to result in different features.

Prostate Morphogenesis, Preliminary Results (2 pages)
Epithelial cells in the prostate are elongated. When they stop communicating, they lose their alignment with other cells and break through the basement membrane. Alignment must be a feature of.

Mathematical Methods (2 pages)
Lattice-based models, LGCA template-based models continuum to discrete and discrete to continuum and embracing a gray area (you don’t have a perfect description in either case due to non-physicality or due to computational limits. Even if you could compute the interaction of every atom, such an approach does not lend further understanfing – as reality is exactly such a computation.)

Teaching Philosophy: Need for critical thinking and an understanding of the world as something that can be understood ( 2 pages)
To some extent, the idea of mathematics as esoteric and Platonic does not help. If mathematics exists in a separate divine realm, how can a student expect to access it for success? I do not advocate the beauty of mathematics. These attract types that would be attracted to mathematics anyway. Instead, the world provides plenty of opportunity where solving problems is required, and my goal is to focus on these problems and how solutions aren’t found in a parallel plane of existence but in the real mechanisms that cause the problems in the first place. Cancer is cells. How do cells behave and how do they organize, given that they are individuals? modeling course : would like to develop in detail a model of ferromagnetism, this requires elongation of particles! would like to enlist a graduate student for this project as it would not be terribly difficult teaching in mathematics
 elementary course
 conference for elementary teachers